פעילויות השבוע

Chung and Li [Invent. Math. 2015] proved that for every expansive action of a countable polycyclic-by-finite group $\Gamma$ on a compact group $X$ by continuous group automorphisms, positive entropy implies the existence of non-diagonal asymptotic pairs. In the same paper they asked if the this holds in general for an expansive action of a countable amenable group $\Gamma$ on a compact space $X$.

In my talk I plan to explain the notions involved Chung and Li‘s question and discuss a property of dynamical systems called the ``pseudo-orbit tracing property‘‘. R. Bowen introduced the pseudo-orbit tracing property in the 1970‘s for $\mathbb{Z}$-actions while studying Axiom A maps. I will prove that Chung and Li‘s question has an affirmative answer if one also assumes pseudo-orbit tracing, and explain implications for algebraic actions (automorphisms of compact abelian groups).

I will also explain why the answer to Chung and Li‘s question is negative if one doesn‘t assume the pseudo-orbit tracing property, even when the acting group is $\mathbb{Z}$, or when the action is algebraic (but not both).

A linearly ordered structure is weakly o-minimal if every definable set is a finite boolean combination of convex sets. A weakly o-minimal expansion of an ordered group is non-valuational if it admits no non-trivial definable convex sub-groups. By a theorem of Baizalov-Poizat if M is an o-minimal expansion of a group and N is a dense elementary substructure then the structure induced on N by all M-definable sets is weakly o-minimal non-valuational.

It is natural to ask whether all non-valuational structures are obtained in this way. We will give examples showing that this is not the case. We will show, however, that if M is non-valuational then there exists M^, an o-minimal structure embedding M densely (as an ordered set) such that M (as a pure set) extended by all M^-definable sets is precisely the structrue M. We will give a complete axiomatisation of the theory of the pair (M^,M), show that it depends only on the theory of M, and that it shares many common features with the theory of dense o-minimal pairs. In particular (M^,M) has dense open core (i.e., the reduct consisting only of definable open sets is o-minimal).

Based on joint work with E. Bar-Yehuda and Y. Peterzil.

A Ramanujan graph is a finite graph which behaves, in terms of expansion, like its universal cover (which is an infinite tree). In recent years a parallel theory has emerged for simplicial complexes of higher dimension, where the role of the tree is taken by Bruhat-Tits buildings. I will recall briefly the story of Ramanujan graphs, and then explain what are ”Ramanujan complexes“, and survey some of the new results regarding their construction and their properties.

Actions of countable discrete groups $\Gamma$ on a compact (metrizable) group $X$ by (continuous) group automorphisms are a rich class of dynamical systems. The case where $X$ is abelian is an important subclass, also called ”algeraic actions“. By Pontryagin duality, algebraic actions are in one-to-one correspondence with $\mathbb{Z}\Gamma$-modules. There is a fascinating ”dictionary“ between the two, a beautiful interplay between dynamics, Fourier analysis, and commutative or noncommutative algebra. In the last several years, much progress has been made towards understanding the algebraic actions of general countable groups. Somehow surprisingly, operator algebras turn out to be important for such a study. This introductory talk will cover some basic aspects of the theory. (New results and open questions might be discussed in a followup talk).